# What is the effect of increasing the size interval in the generalized dynamic equation (GDE) to simulate an evolving aerosol?

The following differential equation allows me to calculate the the number concentration of aerosol particles from 2 nm to 3 nm.

$$\frac{dN_2{_-{_3}}}{dt}=J_2-J_3-N_2{_-{_3}}.CoagS_2{_-{_3}}$$

In the above equation, $$\frac{dN_2{_-{_3}}}{dt}$$ is the rate of change of number concentration of the aerosol particles in $$cm^-{^3}s^-{^1}$$, $$J_2$$ is the formation rate of atmospheric particles at 2 nm in $$cm^-{^3}s^-{^1}$$, $$J_3$$ is the formation rate of atmospheric particles at 3 nm in $$cm^-{^3}s^-{^1}$$, $$N_2{_-{_3}}$$ is the number concentration of particles in $$cm^-{^3}$$ and $$CoagS_2{_-{_3}}$$ is the coagulstion sink of the aerosol particles from 2-3 nm in $$s^-{^1}$$.

If we assume a pseudo steady-state between the sources and sinks of 2-3 nm atmospheric molecular clusters, then: $$\frac{dN_2{_-{_3}}}{dt}=0$$

A consequence of this is:

$$N_2{_-{_3}}=\frac{J_2-J_3-N_2{_-{_3}}}{CoagS_2{_-{_3}}}$$

Number concentration of aerosol particles due to primary emissions are known to increase at larger aerosol particle sizes. Therefore, (Kulmala et al., 2021) states that by using a higher value of 25 nm instead of 3 nm as the upper interval bound $$d_2$$, which is the upper interval bound aerosol particle diameter in nm, for the calculation of $$J_2$$ and $$J_3$$, the influence of primary emissions on $$N_2{_-{_3}}$$ are reduced. The apparent logic of taking a larger size interval for the calculation of $$J_2{_-{_3}}$$ is that the increase in the primary emissions at larger sizes are offsetting the formation rate at both the lower interval bound $$d_1$$ and the upper interval bound $$d_2$$. Although, in terms of absolute numbers, the calculated values of $$J_2$$ and $$J_3$$ would be less accurate using this larger size interval. Mathematically:

$$N_2{_-{_3}}=N_2{_-{_2{_5}}}-N_3{_-{_2{_5}}}$$

Could someone explain to me, mathematically, how is choosing a higher value of 25 nm instead of 3 nm as the upper interval bound $$d_2$$ to calculate $$N_2{_-{_3}}$$, offsetting the formation rate at both the lower interval bound $$d_1$$, 2 nm, and the upper interval bound $$d_2$$, 3 nm?